Problem: Solve for $x$ and $y$ using elimination. $\begin{align*}-x-2y &= 1 \\ 7x+9y &= 3\end{align*}$
Answer: We can eliminate $x$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $7$ and the bottom equation by $1$ $\begin{align*}-7x-14y &= 7\\ 7x+9y &= 3\end{align*}$ Add the top and bottom equations. $-5y = 10$ Divide both sides by $-5$ and reduce as necessary. $y = -2$ Substitute $-2$ for $y$ in the top equation. $-x-2( -2) = 1$ $-x+4 = 1$ $-x = -3$ $x = 3$ The solution is $\enspace x = 3, \enspace y = -2$.